Complex numbers

Simplify the complex numbers in term of i.

z= (1+3i)/(1-4i)

Expert Solution

Multiply numerator and denominator by the complex conjugate of the denominator (1+4i).

(1+3i)/(1-4i) ×(1+4i)/(1+4i)

=[ (1+3i)×(1+4i)] /[(1)^2-(4i)^2]

=(1+7i+12i^2)/(1-16i^2)

Take. i^2=-1

= (1+7i+12(-1))/(1-16(-1))

=(7i-11)/17

Or z= 7i/17-11/17

After simplify we will get

z=7/17i-11/17

Nitish Mishra answered 2 years ago

Find all the complex numbers z for which the multiple-valued function (1+i)^z provides complex numbers which...

Find all the complex numbers z for which the multiple-valued function (1+i)^z provides complex numbers which all have the same absolute value.

Write minimal code to declare a class Complex that enables operations on complex numbers, and also...

Write minimal code to declare a class Complex that enables operations on complex numbers, and also to overload the multiplication * operator.

(a) look at these the complex numbers z1 = − √ 3 + i and z2...

(a) look at these the complex numbers z1 = − √ 3 + i and z2 = 3cis(π/4). write the following complex numbers in polar form, writing your answers in principal argument: i. z1 ii. z1/|z1|. Additionally, convert only this answer into Cartesian form. iii. z1z2 iv. z2/z1 v. (z1) -3 vi. All complex numbers w that satisfy w 3 = z1. (b) On an Argand diagram, sketch the subset S of the complex plane defined by S = {z...

Construct the field of complex numbers as an extension of the field of rationales. Please, be...

Construct the field of complex numbers as an extension of the field of rationales. Please, be thorough.

How to construct a field of complex numbers as an extension of the field of rationales?

Write the following complex numbers in polar form, as ??^??. 1/3+4i

Complex Numbers: What's the difference between Log(z), log(z) and ln(z)? where z is a complex number,...

Complex Numbers: What's the difference between Log(z), log(z) and ln(z)? where z is a complex number, z = x + iy

Due October 25. Let R denote the set of complex numbers of the form a +...

Due October 25. Let R denote the set of complex numbers of the form a + b √ 3i, with a, b ∈ Z. Define N : R → Z≥0, by N(a + b √ 3i) = a 2 + 3b 2 . Prove: (i) R is closed under addition and multiplication. Conclude R is a ring and also an integral domain. (ii) Prove N(xy) = N(x)N(y), for all x, y ∈ R. (ii) Prove that 1, −1 are the...

Logic & Sets (Proofs question) Show that complex numbers cannot be ordered in a way that...

Logic & Sets (Proofs question) Show that complex numbers cannot be ordered in a way that satisfies our axioms. Axioms for order: 1. if x is less than/equal to y and w is greater than zero, then wx is less than/equal to wy 2. for w, x, y, z w is less than/equal to x, y is less than/equal to z then w + y = x + z if and only iff w = x and y = z

There is an engineering project that uses complex numbers, electrical circuits. you need to develop a...

There is an engineering project that uses complex numbers, electrical circuits. you need to develop a user-defined structure type and a set of operations that will make complex arithmetic virtually as straightforward as arithmetic on C’s built-in numeric types. You will need to define functions for complex I/O as well as for the basic arithmetic operations (addition, subtraction, and for finding the absolute value of a complex number). Hint: the complex number a + bi has a real part a...

Question